\(\hat{p}\) inherently has some standard error, so when stating an estimate for a population proportion, it is best practice to provide a plausible range of values instead of just the point estimate.

Using only a point estimate is like fishing with a spear - we can throw a spear where we saw a fish, but we will probability miss. A confidence interval, however, is like fishing with a net; it provides a range of plausible values where we are likely to find the population parameter.

The sample proportion \(\hat{p}\) is the most plausible value of the population proportion. the standard error provides a guide for how large we should make the confidence interval.

When the Central Limit Theorem conditions are satisfied, the point estimate closely follows a normal distribution. In a normal distribution, 95% of data points are within 1.96 standard deviations of the mean. Using this principle, we can construct a confidence interval that extends 1.96 standard errors from the sample proportion to be 95% confident that the interval captures the population proportion.

point estimate \(\pm 1.96 \cdot SE\) \(\hat{p} \pm 1.96 \cdot \sqrt{\frac{p(1-p)}{n}}\)

"95% confident" means that if we took many samples and built a 95% confidence interval from each, about 95% of those intervals would contain the parameter, \(p\).

[!def] 95% Confidence Interval for a Parameter When the distribution of a point estimate qualifies for the Central Limit Theorem and therefore closely follows a normal distribution, we can construct a 95% confidence interval as \(\text{point estimate }\pm 1.95 \cdot SE\) (SE = standard error)

[!example] 5.8. A Pew research poll found that 88.7% of a random sample of 1000 Americans supported expanding the role of solar power. Compute and interpret a 95% confidence interval for he population proportion. The standard error is 0.010. \(\(\begin{align} &\hat{p} \pm 1.96 \cdot SE_{\hat{p}} \\ &0.887 \pm 1.96 \cdot 0.010 \\ & (0.8674, 0.9066) \end{align}\)\) We are 95% confident that the actual proportion of Americans who support expanding solar power is betwen 86.7% and 90.7%

Changing the Confidence Interval

If \(X\) is a normally distributed random variable, what is the probability of the value \(X\) being within 2.58 standad deviations of the mean?

This is equivalent to asking how often the Z-score \(z\) will be \(-2.58 < z < 2.58\). We can use statistical software, a calculator, or a table to look up these values for a normal distribution: 0.0049 and 0.9951. Thus, there is a \(0.9951 - 0.0049 \approx 0.99\) probability that an unobserved normal random variable \(X\) will be within 2.58 standard deviations of \(\mu\).

This means that 99% of the time, a normal random variable will be within 2.58 standard deviations of the mean. to create a 99% confidence interva, we change 1.96 in the 95% confidence interval formula to 2.58. Therefore:

[!formula] Formula for 99% confidence interval \(\text{point estimate } \pm 2.58 \cdot SE\)

What about when our model does not fit the normal distribution?

[!def] Confidence interval using any confidence level If a point estimate closely follows a normal model with standard error SE, then a confidence interval for the population parameter is \(\text{point estimate } \pm z^* \cdot SE\) where \(z^*\) corresponds to the confidence level selected.

[!def] Margin of error In a confidence interval, \(z^* \cdot SE\) is called the margin of error.

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[!checklist] Confidence interval for a single proportion 1. Prepare - Identify \(\hat{p}\) and \(n\) and determine the confidence level you want to use. 2. Check - Verify the conditions to ensure \(\hat{p}\) is normal. For one-proportion confidence intervals, use \(\hat{p}\) in place of \(p\) to check the success-failure condition. 3. Calculate - If the conditions hold, compute \(SE\) using \(\hat{p}\), find \(z^*\), and construct the interval. 4. Conclude - Interpret the confidence interval in the context of the problem

Interpreting Confidence Intervals

Confidence intervals are only about the population parameter. It says nothing about individual observations or point estimates.